Lemma 86.26.3. Let $A \to B$ be a local map of local Noetherian rings such that

1. $A \to B$ is flat,

2. $\mathfrak m_ B = \mathfrak m_ A B$, and

3. $\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$

Then the base change functor from the category (86.26.0.1) for $(A, \mathfrak m_ A)$ to the category (86.26.0.1) for $(B, \mathfrak m_ B)$ is an equivalence.

Proof. The conditions signify that $A \to B$ induces an isomorphism on completions, see More on Algebra, Lemma 15.43.9. Hence this lemma is a special case of Lemma 86.26.1. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).